Abstract In a recent work the first named author, Levitin and Vassiliev have constructed the wave propagator on a closed Riemannian manifold M as a single oscillatory integral global both in space and in time with a distinguished complex-valued phase function. In this paper, first we give a natural reinterpretation of the underlying algorithmic construction in the language of ultrastatic Lorentzian manifolds. Subsequently we show that the construction carries over to the case of static backgrounds thanks to a suitable reduction to the ultrastatic scenario. Finally we prove that the overall procedure can be generalised to any globally hyperbolic spacetime with compact Cauchy surfaces. As an application, we discuss how, from our procedure, one can recover the local Hadamard expansion which plays a key role in all applications in quantum field theory on curved backgrounds.

中文翻译：

---

全局双曲时空上的全局波参数

摘要 在最近的一项工作中，第一作者列维京和瓦西里耶夫构建了封闭黎曼流形 M 上的波传播子，作为一个在空间和时间上均具有显着复值相位函数的全局单一振荡积分。在本文中，我们首先用超静态洛伦兹流形语言对底层算法构造进行自然的重新解释。随后，我们展示了由于对超静态场景的适当减少，该构造可以延续到静态背景的情况。最后我们证明整个过程可以推广到任何具有紧凑柯西表面的全局双曲时空。作为一个应用程序，我们讨论如何从我们的程序中，